Which of the following is a pair of co-primes?

### Concept & Logic Two numbers are considered co-prime (or mutually prime) if they share no common factors other than $1$. In other words, their Highest Common Factor (HCF) must be exactly $1$: $$HCF(a, b) = 1$$ ### Step-by-Step Solution Let's analyze the factors of each pair given in the options to find their HCF: * **Option (a) (16, 62):** Both numbers are even. Therefore, they are at least divisible by $2$. The HCF is not $1$. * **Option (c) (21, 35):** Both numbers are multiples of $7$ ($7 \times 3 = 21$ and $7 \times 5 = 35$). The HCF is $7$. * **Option (d) (23, 92):** $23$ is a prime number. If we multiply $23 \times 4$, we get $92$. Thus, $23$ is a factor of $92$, making their HCF $23$. * **Option (b) (18, 25):** The prime factorization of $18$ is $2 \times 3^2$. The prime factorization of $25$ is $5^2$. They share absolutely no common prime factors. Their HCF is $1$. ### Exam Strategy & Shortcut Use divisibility rules to quickly eliminate options. Any pair of even numbers (like 16 and 62) can be instantly crossed out. Check for obvious multiples (21 and 35 are clearly in the 7 times table). This leaves (18, 25) as the obvious answer within seconds. ### Common Pitfall Students often confuse "co-prime" with "prime numbers." Neither $18$ nor $25$ are prime numbers individually, but they are co-prime *to each other* because they share no common factors. Do not look for pairs where both numbers are prime; look for pairs with no common overlap. ### Final Answer **Therefore, the correct answer is (18, 25).**